Blow-up results and localization of blow-up points in an \(N\)-dimensional smooth domain
type: Item
instance of: scholarly article
title: Blow-up results and localization of blow-up points in an \(N\)-dimensional smooth domain
zbMATH Open document ID: 0884.35071
DOI: 10.1215/S0012-7094-97-08816-5
author: Diego F. Rial
author: J. D. Rossi
published in: Duke Mathematical Journal
publication date: 1998-04-02T00:00:00Z
review text: Positive solutions of the heat equation are studied, on a bounded and smooth cylindrical domain \(\Omega\times ]0,T[\), subject to the conditions \(D_\eta u=f(u)\) on \(\partial\Omega \times ]0,T[\) and \(u(\cdot,0) =u_0\) on \(\Omega\). Here \(f\) is an increasing, positive, \(C^2\)-function such that \(\int^\infty (1/f) <\infty\), while \(u_0\) is a positive function in \(C^{2+\alpha} (\overline \Omega)\) such that \(D_\eta u_0 =f(u_0)\). An estimate for the blow-up time is established. Under the additional hypotheses that \(\Omega= B(0,1)\), \(f(0)=0\), and \(f\) is convex, it is also proved that the blow-up occurs at the boundary. For a general \(\Omega\), the same conclusion is shown to hold if the hypothesis \(\Delta u_0\geq c>0\) is also assumed. These results generalize earlier ones of several authors.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1042489
zbMATH Keywords: blow-up at the boundary
zbMATH Keywords: nonlinear boundary condition
zbMATH Keywords: positive solutions
zbMATH Keywords: heat equation
zbMATH Keywords: estimate for the blow-up time
reviewed by: Neil A. Watson
MaRDI profile type: Publication
cites work: The Profile Near Blowup Time for Solution of the Heat Equation with a Nonlinear Boundary Condition
cites work: Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time
cites work: Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition
cites work: On Existence and Nonexistence in the Large of Solutions of Parabolic Differential Equations with a Nonlinear Boundary Condition
cites work: Global Existence and Blow-Up Problems for Quasilinear Parabolic Equations with Nonlinear Boundary Conditions
cites work: Global Behavior of Positive Solutions to Nonlinear Diffusion Problems with Nonlinear Absorption Through the Boundary
full work available at URL: s0012-7094-97-08816-5
OpenAlex ID: W2081407931
description: scientific article; zbMATH DE number 1042489